Can Isotropic Thermal Conduction Heat One of the Hottest Cool Cores?

Thermal conduction can act as a heating mechanism for cool-core clusters, yet the same transport must erode any temperature discontinuity it crosses. Ophiuchus permits both effects to be tested in the same atmosphere. Its temperature rises from about 1 keV in the innermost core to 9 keV at r about30 kpc, making conductive heating especially favorable, while two sharply resolved cold fronts independently constrain the same transport. We construct a steady radial energy balance from deprojected Chandra density and temperature profiles, with XRISM-based limits on turbulent heating and a subsonic inflow contribution. Closing the balance with conduction requires only sub-Spitzer coefficients --- at 25 kpc and and at the inner and outer cold fronts --- so conduction can energetically supply the missing heat. Yet the same coefficients, applied normal to the cold-front surfaces, would broaden the fronts to their observed width limits within and Myr, short compared to the characteristic sloshing timescales that generate and sustain such interfaces. For a representative 200 Myr residence time, the radial conductivity demanded by the energy balance exceeds the cross-front conductivity permitted by front survival by roughly a factor of three, with the required anisotropy exceeding unity in 99.9% and 98.8% of accepted models. The result leaves two broad regimes. Thermal conduction is either subdominant in Ophiuchus or contributes substantially but is strongly anisotropic, with heat transport suppressed across the cold fronts relative to the radial direction. Together they disfavor locally isotropic conduction as a dominant heating mechanism in cool cores generally.